Star configuration points and generic plane curves
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l 1 comma ellipsis comma script l Subscript l Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>l</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell _1,\ldots ,\ell _l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lines in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that no three lines meet in a point. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X left-parenthesis l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}(l)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet script l Subscript i Baseline intersection script l Subscript j Baseline vertical-bar 1 less-than-or-equal-to i greater-than j less-than-or-equal-to l EndSet subset-of-or-equal-to double-struck upper P squared"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> ∩ </mml:mo> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>j</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>l</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo> ⊆ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\ell _i \cap \ell _j ~|~ 1 \leq i > j \leq l\} \subseteq \mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We call <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X left-parenthesis l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {X}(l)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a star configuration. We describe all pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d comma l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d,l)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the generic degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> curve in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X left-parenthesis l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mro
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it